# Energy in Simple Harmonic Motion (SHM)

This link is from a book

It’s well worth a look or two, if only to illustrate the extent of some of these applications in physics and also the fact that SH oscillators are nothing more than periodic exchangers of KE and PE.

Since KE is proportional to v² the frequency of the KE/time graph is twice that of the v/t graph. The same applies to the PE /time graph, but it is inverted. The graphs tell the story

if

then kinetic energy against time is a cosine squared function and so potential energy against time is a sine squared function, both oscillating at twice the frequency of the displacement. Think this through carefully by imagining either a pendulum or a mass-spring oscillator during various points in its cycle and remember, energy is always positive.

For a perfect oscillator, the sum of the energies at any point in the cycle is constant, whether we plot energy against time or energy against displacement as shown in the document. Download a copy and print it for your notes. Here’s the graph – it’s parabolic, clearly because of the squared term in v.

For energy against time:

and energy against displacement

HL IB students: you should be able to use these equations. Don’t attempt to measure gradients of graphs unless explicitly asked to do so. This way is better. Starting with this

showing us that when x=r, at the extremities of the oscillation, v=0, as we’d expect. Replacing r with  x0 we can find values for velocity kinetic energy and potential energy for any value of displacement.